This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. Primary support for the subproject and the subproject's principal investigator may have been provided by other sources, including other NIH sources. The Total Cost listed for the subproject likely represents the estimated amount of Center infrastructure utilized by the subproject, not direct funding provided by the NCRR grant to the subproject or subproject staff. The diagonalization of the stochastic Liouville (SL) matrix using the Lanczos algorithm (LA) is optimized with the aid of the conjugate (CG) method for calculating 2D-ELDOR spectra in the slow-motional regime. In each step of the LA recursion, the convergence is monitored according to the residual norm calculated in the CG iterations. Thus the methods of the CG and LA can be coupled together to tri-diagonalize a large symmetric and complex sparse matrix efficiently. The LA-CG has been very successfully used to tri-diagonalize the SLE matrix in the slow-motional regime. However, due to the loss of orthogonality in the LA vectors, we found that the LA-CG method can break down in calculating spectra in the very slow-motional regime. This is mainly due to the fact that CG requires the residual norm to calculate the solution direction in each iteration. This indicates that the whole algorithm would be spoiled due to the loss of orthogonality after a certain number of LA projections. This would be particularly true when one calculates a very slow-motional spectrum that requires a large number of LA projections in order to adequately obtain good eigenvalues. The quasi-minimal residual (QMR), which was originally developed to be a linear equation solver, has been adapted for determining the number of LA projections in our program. In the QMR method, the solution vector is obtained by minimizing the quasi-residual norm, using QR factorization. It is of great advantage to us to replace CG with QMR to determine the number of LA projections, because the residual norm is NOT used iteratively as in the CG method to determine the solution direction in each iteration. Therefore, this LA-QMR method that we utilize provides an advantage of avoiding the breakdown noted above.